The Gromov width of $4$–dimensional tori

Abstract

Let $\omega$ be any linear symplectic form on the $4$–torus $T^4$. We show that in all cases $\left(T^4,\omega \right)$ can be fully filled by one symplectic ball. If $\left(T^4,\omega \right)$ is not symplectomorphic to a product $T^2\left(\mu \right)\times T^2\left(\mu \right)$ of equal sized factors, then it can also be fully filled by any finite collection of balls provided only that their total volume is less than that of $\left(T^4,\omega \right)$.